Journal of Automated Reasoning

, Volume 21, Issue 3, pp 357–380 | Cite as

A New Approach for Automatic Theorem Proving in Real Geometry

  • Andreas Dolzmann
  • Thomas Sturm
  • Volker Weispfenning


We present a new method for proving geometric theorems in the real plane or higher dimension. The method is derived from elimination set ideas for quantifier elimination in linear and quadratic formulas over the reals. In contrast to other approaches, our method can also prove theorems whose complex analogues fail. Moreover, the problem formulation may involve order inequalities. After specification of independent variables, nondegeneracy conditions are generated automatically. Moreover, when trying to prove conjectures that – apart from nondegeneracy conditions – do not hold in the claimed generality, missing premises are found automatically. We demonstrate the applicability of our method to nontrivial examples.

real quantifier elimination real geometry automatic theorem proving over the reals 


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  1. 1.
    Abdallah, C. T., Dorato, P., Liska, R., Steinberg, S., and Yang, W.: Applications of quantifier elimination theory to control system design, in 4th IEEE Mediterranean Symposium on Control and Automation, IEEE, 1996.Google Scholar
  2. 2.
    Arnon, D. S.: A bibliography of quantifier elimination for real closed fields, J. Symbolic Comput. 5(1–2) (1988), 267–274.Google Scholar
  3. 3.
    Basu, S., Pollack, R., and Roy, M.-R.: On the combinatorial and algebraic complexity of quantifier elimination, in S. Goldwasser (ed.), Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Los Alamitos, CA, USA, November 1994, IEEE Computer Society Press, 1994, pp. 632–641.Google Scholar
  4. 4.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Doctoral dissertation, Mathematical Institute, University of Innsbruck, Innsbruck, Austria, 1965.Google Scholar
  5. 5.
    Buchberger, B.: Applications of Gröbner bases in non-linear computational geometry, in R. Janβen (ed.), Trends in Computer Algebra, Proceedings, Lecture Notes in Comput. Sci. 296, Springer, 1988, pp. 52–80.Google Scholar
  6. 6.
    Carrá-Ferro, G. and Gallo, G.: A procedure to prove geometrical statements, Technical report, Dip. Mathematica Univ. Catania, Italy, 1987.Google Scholar
  7. 7.
    Chou, S.-C.: Mechanical Geometry Theorem Proving, Mathematics and Its Applications, D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1988.Google Scholar
  8. 8.
    Collins, G. E. and Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination, J. Symbolic Comput. 12(3) (1991), 299–328.Google Scholar
  9. 9.
    Colmerauer, A.: Prolog III, Communications of the ACM 33(7) (1990), 70–90.Google Scholar
  10. 10.
    Corless, R. M. and Jeffrey, D. J.: Well... it isn't quite that simple, ACM SIGSAM Bulletin 26(3) (1992), 2–6. Feature.Google Scholar
  11. 11.
    Dolzmann, A. and Sturm, T.: Redlog user manual, Technical Report MIP-9616, FMI, Universität Passau, D-94030 Passau, Germany, October 1996. Edition 1.0 for Version 1.0.Google Scholar
  12. 12.
    Dolzmann, A. and Sturm, T.: Guarded expressions in practice, in W. W. Küchlin (ed.), Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (ISSAC 97), ACM Press, New York, 1997, pp. 376–383.Google Scholar
  13. 13.
    Dolzmann, A. and Sturm, T.: Redlog: Computer algebra meets computer logic, ACM SIGSAM Bulletin 31(2) (1997), 2–9.Google Scholar
  14. 14.
    Dolzmann, A. and Sturm, T.: Simplification of quantifier-free formulae over ordered fields, J. Symbolic Comput. 24(2) (1997), 209–231.Google Scholar
  15. 15.
    Hong, H., Collins, G. E., Johnson, J. R., and Encarnacion, M. J.: QEPCAD interactive version 12. Kindly communicated to us by Hoon Hong, September 1993.Google Scholar
  16. 16.
    Hong, H., Liska, R., and Steinberg, S.: Testing stability by quantifier elimination, J. Symbolic Comput. 24(2) (1997), 161–187. Special issue on applications of quantifier elimination.Google Scholar
  17. 17.
    Kapur, D.: Using Gröbner bases to reason about geometry problems, J. Symbolic Comput. 2(4) (1986), 399–408.Google Scholar
  18. 18.
    Kutzler, B. A.: Algebraic Approaches to Automated Theorem Proving, PhD Thesis, Johannes Kepler Universität Linz, 1988. RISC-Linz series no. 88-74.0.Google Scholar
  19. 19.
    Kutzler, B. A. and Stifter, S.: On the application of Buchberger's algorithm to automated geometry theorem proving, J. Symbolic Comput. 2(4) (1986), 389–397.Google Scholar
  20. 20.
    Loos, R. and Weispfenning, V.: Applying linear quantifier elimination, Comput. J. 36(5) (1993), 450–462. Special issue on computational quantifier elimination.Google Scholar
  21. 21.
    MacLane, S.: Some interpretations of abstract linear dependence in terms of projective geometry, Amer. J. Math. 58 (1936), 236–240.Google Scholar
  22. 22.
    McPhee, N. F., Chou, S.-C., and Gao, X.-S.: Mechanically proving geometry theorems using a combination of Wu's method and Collins' method, in Alan Bundy (ed.), Automated Deduction – CADE-12, Lecture Notes in Artif. Intell. 814, Springer, Berlin, Heidelberg, New York, 1994, pp. 401–415.Google Scholar
  23. 23.
    Preparata, F. P. and Shamos, M. I.: Computational Geometry – An Introduction, Texts and Monographs in Computer Science, Springer, New York, 1985.Google Scholar
  24. 24.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, Part I–III, J. Symbolic Comput. 13(3) (1992), 255–352.Google Scholar
  25. 25.
    Seidenberg, A.: An elimination theory for differential algebra, Univ. California Publ. Math. (N.S.) 3 (1956), 31–66.Google Scholar
  26. 26.
    Seidenberg, A.: Some remarks on Hilbert's Nullstellensatz, Arch. Math. 7 (1956), 235–240.Google Scholar
  27. 27.
    Wang, D.-M.: An elimination method for polynomial systems, J. Symbolic Comput. 16(2) (1993), 83–114.Google Scholar
  28. 28.
    Wang, D.-M.: Reasoning about geometric problems using an elimination method, in J. Pfalzgraf (ed.), Automatic Practical Reasoning, Springer-Verlag, Wien, 1995, pp. 147–185.Google Scholar
  29. 29.
    Wang, D.-M. and Gao, X.-S.: Geometry theorems proved mechanically using Wu's method – part on Euclidean geometry, Mathematics-Mechanization Research Preprints 2, Institute of Systems Science, Academia Sinica, Beijing, China, November 1987.Google Scholar
  30. 30.
    Weispfenning, V.: The complexity of linear problems in fields, J. Symbolic Comput. 5(1) (1988), 3–27.Google Scholar
  31. 31.
    Weispfenning, V.: Parametric linear and quadratic optimization by elimination, Technical Report MIP-9404, FMI, Universität Passau, D-94030 Passau, Germany, April 1994. To appear in the J. Symbolic Comput. Google Scholar
  32. 32.
    Weispfenning, V.: Quantifier elimination for real algebra – the cubic case, in Proceedings of the International Symposium on Symbolic and Algebraic Computation in Oxford, ACM Press, New York, 1994, pp. 258–263.Google Scholar
  33. 33.
    Weispfenning, V.: Quantifier elimination for real algebra – the quadratic case and beyond, Appl. Algebra Eng. Comm. Comput. 8(2) (1997), 85–101.Google Scholar
  34. 34.
    Weispfenning, V.: Simulation and optimization by quantifier elimination, J. Symbolic Comput. 24(2) (1997), 189–208. Special issue on applications of quantifier elimination.Google Scholar
  35. 35.
    Winkler, F.: A geometrical decision algorithm based on the Gröbner bases algorithm, in P. Gianni (ed.), Symbolic and Algebraic Computation, Proceedings of ISSAC' 88, Lecture Notes in Comput. Sci. 358, Springer, Berlin, Heidelberg, 1988, pp. 356–363.Google Scholar
  36. 36.
    Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry, Scientia Sinica 21 (1978), 159–172, also Contemporary Mathematics 29 (1984), 213–234.Google Scholar
  37. 37.
    Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometries, J. Systems Sci. Math. Sci. 4 (1984), 207–235.Google Scholar
  38. 38.
    Wu, W.-T.: Some recent advances in mechanical theorem-proving of geometries, Contemporary Mathematics 29 (1984), 235–241.Google Scholar
  39. 39.
    Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometry, J. Automated Reasoning 2 (1986), 219–252.Google Scholar
  40. 40.
    Wu, W.-T.: On problems involving inequalities, Mathematics-Mechanization Research Preprints 7, Institute of Systems Science, Academia Sinica, Beijing, China, March 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Andreas Dolzmann
    • 1
  • Thomas Sturm
    • 1
  • Volker Weispfenning
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität PassauGermany

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